Force analysis of the two-satellite planetary mechanism with elliptical gears

Non-circular gears can be used in modern machines and mechanisms for the implementation of various types of motion and have high strength and compactness compared to linkage mechanisms. This article presents the force analysis of non-circular gear on the example of the planetarymechanismwith elliptical gears, providing the rotationally reciprocatingmotion of the impeller of the stirred tank. Based on the calculation schemes of the links, kinetostatic balance equations for each link of the mechanism are compiled and solved. Reaction forces in kinematic pairs and balancing moment on the input shaft of the mechanismare found. The results can be used in the synthesis and analysis of various machines with the proposed kinematic scheme of the mechanism.


Introduction
Transmissions by non-circular gears have been known for a long time and are of great interest to scientists and engineers [1,2]. The first schemes of mechanisms with noncircular wheels were found in the works of Leonardo da Vinci, and some of the first models of such devices were made in the 19th century by the German scientist Ferdinand Redtenbacher. Widespread research and dissemination of these gears has long been hindered by the lack of simple and accurate manufacturing methods for non-circular gears.
Currently, there are a large number of papers devoted to geometric and kinematic analysis, design and practical application of gears by non-circular gears [3][4][5][6][7][8][9]. For example, there are proposed: chain drives with variable *Corresponding Author: A. A. Prikhodko: Department of Land Transport and Mechanics, Kuban State Technological University, 350072 Krasnodar, Russia; Email: sannic92@gmail.com ratio for bicycles [3]; a mechanical device containing a Maltese cross mechanism and a gearing to achieve intermittent motion [4]; the gear trains used for velocity variation and function generation [5]; constructions of non-circular gears as a part of rotor hydraulic machines [6] and highperformance pumps [7]; a knee motion assist device for biomechatronic exoskeleton having grooved cams and noncircular wheels [8]; single planetary non-circular gears with one internal and one external gear for bicycles with high efficiency [9]. It was shown in [10][11][12][13][14] that planetary gears with non-circular wheels allow to change the sign of the transfer function and, therefore, provide different types of motion, for example, intermittent motion [10][11][12] or rotationally reciprocating motion [13,14]. The proposed mechanisms are more compact and durable compared to linkages, and the improvement of high-precision equipment for machining and additive production stimulates the development of this issue of the mechanism and machine theory.
Non-circular gears can have various shapes, but elliptical gears are the most researched and widespread at the moment [15][16][17]. A large number of mechanisms and devices based on elliptical gears have been developed, their geometry and kinematics have been investigated [18][19][20], and various problems of their manufacture have been solved [21][22][23].
However, most of the papers on the creation of noncircular gears are devoted only to the geometry and kinematics of these mechanisms, while the issues of their dynamics are much more complex and have not been studied. Nevertheless, some applied problems of the dynamics of non-circular gears have been solved. For example, Xing Liu et al. [24] carried out a theoretical and experimental study of the dynamic performance of elliptical gears with rotational axes at the focus and center of the pitch ellipse. Nan Gao et al. [17] investigated parametric vibrations and instabilities of elliptical gears caused by loading torque and eccentricity vibrations. Zhiqin Cai and Chao Lin [25] presented and investigated a generalized nonlinear dynamic model of a curved gear drive based on Lagrange-Bondon graphs. Many researchers consider the issues of dynamics in relation to practical applications of mechanical devices [24][25][26] containing non-circular gears, since dynamic models include the parameters of technological load on the working body of the machine.
The purpose of this article is the kinetostatic analysis of a planetary mechanism with two external gears operating as part of a drive of a rotationally reciprocating stirred tank. The idea of imparting various non-stationary types of motion, including rotationally reciprocating motion, to the impeller of the stirred tank has been supported by many researchers around the world [27][28][29]. Despite the increasing complexity of the drive constructions, the prospects of this research area are due to a significant increase in the mixing efficiency, which has been experimentally confirmed using various practical examples [30,31].

Formulation of the problem
The object of this research is the actuator of the stirred tank ( Figure 1a), in which the rotational motion of the motor shaft is converted into the reciprocating motion of the impeller due to a planetary mechanism with elliptical gears (Figure 1b). The proposed mechanism (Figure 1b) consists of rack 0, three-vertex links 1, 2, 3, two-vertex link 4, 1-DOF kinematic pairs A, C, D, G and 2-DOF kinematic pairs B, E, F. The structural analysis carried out in [13] showed that the mechanism exists in a three-moving space (translational displacements along the x and y axes and rotation around the z axis), which is traditionally called "planar." In the study of kinetostatics, we assume that all the kinematic pairs lie in the same plane. Figure 2 presents the calculation scheme of the mechanism. The angles of rotation of the input shaft together with the carrier and the output shaft are designated as φ 1 and φ 3 , respectively. The study of kinetostatics was carried out for the actuator of the stirred tank, operating in the following mode: the frequency of the rotationally reciprocating motion f = 10 Hz, the swing angle of the impeller α = 150 ∘ . For this mode, the following calculation data are considered known from [13,14,32]: active forces, moments of forces, masses and moments of inertia of links, as well as kinematic parameters (positions, velocities and accelerations). It is also assumed that the friction forces in the kinematic pairs are equal to zero.

Force analysis of the mechanism
Conducting a force analysis and determining reactions in kinematic pairs consists of writing and solving kinetostatic equations. In the proposed mechanism, for each link, it is necessary to compose 3 equations, in total 12 equations. The number of unknown reactions is also 12: 1 reaction in the kinematic pairs B, E, F and 2 components of the reaction in the kinematic pairs A, C, D, G, and also the balancing Force analysis is carried out sequentially for each link, beginnign with conducting a force analysis from link 4, since when it is released from bonds, the smallest number of unknown reactions is formed. In accordance with the d'Alembert principle, the acting forces and moments, as well as the forces and moments of force inertia were applied to link 4 ( Figure 3). Link 4 makes plane-parallel motion, the instantaneous center of velocity, of which is the point on the sun wheel, opposite point B. Dynamic analysis showed [14,32] that the angular velocity of the input link 1 is not a constant value due to the changes in the load on the impeller. However, when using the flywheel, the speed fluctuations are insignificant and do not have much effect on the forces acting on the links of the machine. Therefore, for the convenience of calculations, we assume that ω 1 = const, ω 2 = const and ω 4 = const.
Link 4 is affected by: a reaction ⃗ R 41 in kinematic pair D, the direction of which is not known in advance; reaction ⃗ R 43 in the kinematic pair E, which is directed at an angle α to the tangent O 1 O 2 ; as well as inertial forces, which are applied to the center of mass of the link. Gravity forces are directed perpendicular to the plane of the figure, therefore, they are not projected on the x and y axes. (Figure 3(5)) make an angle with the axis Ox 2 . Since O 3 O 4 is a tangent to the ellipse at the point F, to determine the angle , the equation of the tangent in the plane Ox 2 y 2 is written as [33,34]: are the coordinates of the point F in the plane Ox 2 y 2 , a and b are the semiaxes of the ellipse, OG = c is the focal distance of the ellipse. Transforming equation (1), we obtain an expression for finding : To determine the inertia forces, link 4 is considered as the sum of two bodies: body 4.1 (counterweight and satellite shaft), whose inertia force ⃗ F in41 is applied to point D, and body 4.2 (elliptical wheel), whose inertia force ⃗ F in42 is applied to point D 1 .
The accelerations of points D and D 1 are defined as follows. Since ε 1 = 0, then a D = a n D = ω 2 1 · AD. Taking point D as a pole, we obtain ⃗ a D1 = ⃗ a D + ⃗ a D1D , where a D1D = a n D1D = ω 2 2 · D 1 D is the acceleration of point D 1 relative to pole D.
The required forces of inertia ⃗ F in41 and ⃗ F in42 are determined as: The notations ⃗ F 1 = m 42 · ⃗ a D , ⃗ F 2 = m 42 · ⃗ a D1D is introduced ( Figure 3) and the equilibrium condition is written in the form of a system of three equations -the sum of the projections of all the forces on the x and y axis, and the sum of the moments relative to the point D: − R 43 · sin ( + α − φ 3) = 0; Considering that R x 43 = R 43 · cos ( + α − φ 3) and R y 43 = R 43 · sin ( + α − φ 3) , from (4), the reaction R 43 is determined. Then, from equations (2) and (3), we find the reactions R x 41 and R y 41 . Figure 4 shows the kinetostatic equilibrium of link 3. The link makes a rotationally reciprocating motion and is affected by: reaction ⃗ R 30 in the kinematic pair G, the direction of which is not known in advance; reactions ⃗ R 32 and ⃗ R 34 in kinematic pairs F and E directed at an angle α to the tangents O 3 O 4 and O 1 O 2 ; normal ⃗ F n in3 and tangential ⃗ F τ in3 inertia forces applied to the centers of mass of elliptical gears; moment of resistance of the stirred liquid ⃗ Mr and moment of inertia forces ⃗ M in3 . The kinetostatic balance condition is represented as a system of three equations -the sum of the projections of the all the forces on the x and y axis, and the sum of the moments relative to the point G: − R y 32 · FG · sin φ 1 − R x 34 · EG · cos φ 1 − R y 34 · EG · sin φ 1 = 0.
Considering that R x 32 = R 32 · cos( + α − φ 3 ), R y 32 = R 32 · sin( + α − φ 3 ), R x 34 = R 34 · cos( + α − φ 3 ), R y 34 = R 34 · sin( + α − φ 3 ), from equation (7), the reaction R 32 is determined. Then, from equations (5) and (6), we find the reactions R x 30 and R y 30 . Figure 5 presents the kinetostatic equilibrium of link 2, which performs the plane-parallel motion, the instantaneous center of velocity of which is point B. The link is affected by: a reaction ⃗ R 21 in kinematic pair C, the direction of which is not known in advance; reaction ⃗ R 23 in kinematic pair F, which is directed opposite to the found reaction ⃗ R 32 ; reaction ⃗ R 20 in kinematic pair B, which is directed at an angle α to the tangent O 5 O 6 ; as well as inertia forces, which are applied to the center of mass of the link. For the convenience of solving the problem, as in the case of link 4, we consider link 2 as the sum of two bodies: body 2.1 (cylindrical wheel and satellite shaft), the inertia force ⃗ F in21 of which is applied to point C, and body 2.2 (elliptical wheel), the inertia force ⃗ F in22 of which is attached to the point C 1 . The accelerations of points C and C 1 are defined as follows. Since ε 1 = 0, then a C = a n C = ω 2 1 · AC. Taking point C as a pole, we obtain ⃗ a C1 = ⃗ a C + ⃗ a C1 , where a C1 = a n C1 = ω 2 2 · C 1 is the acceleration of point C 1 relative to pole C. The required forces of inertia ⃗ F in21 and ⃗ F in22 are determined as: Figure 5) and the equilibrium condition is written in the form of a system of three equations -the sum of the projections of all the forces on the x and y axis, and the sum of the moments relative to the point C: − R 23 cos ( + α − φ 3) + R 20 cos (α − φ 1) = 0; −R x 23 CF · cos(π + φ 1 ) − R y 23 CF sin(π + φ 1 ) +R x 20 BC cos(π + φ 1 ) + R y 20 BC sin(π + φ 1 ) = 0. (10), the reaction R 20 is determined. Then, from equations (8) and (9), we find the reactions R x 21 and R y 21 . Figure 6 shows the kinetostatic equilibrium of link 1, which performs a rotational motion. The link is affected by: a reaction ⃗ R 10 in kinematic pair A, the direction and module of which are unknown; a reaction ⃗ R 12 in kinematic pair C, which is directed opposite to the found reaction ⃗ R 21 ; reaction ⃗ R 14 in kinematic pair D, which is directed opposite to the found reaction ⃗ R 41 ; balancing moment ⃗ M b . The kinetostatic balance condition is represented as a system of three equations -the sum of the projections of all the forces on the x and y axis, and the sum of the moments relative to the point A: R y 10 − R y 12 − R y 14 = 0; − My + R x 12 · AC · cos φ 1 − R y 12 · AC · sin φ 1 − R x 14 · AD · cos φ 1 + R y 14 · AD · sin φ 1 = 0.

Results
Mathematical modeling of kinetostatics is carried out for the rotationally reciprocating stirred tank with planetary actuator, developed on the basis of the Ika Magic Plant laboratory scale process plant (Figure 7). The developed experimental setup contains an electric motor 1, a motor control unit 2, a rack 3, on which the motor is fixed, a planetary actuator 4, an input shaft of an actuator 5, an output   shaft of an actuator 6, a coupling 7, a reactor 8, a rack 9, which holds the reactor with the actuator, reactor cover 10, sealing clamp 11, impeller seal 12, impeller 13, product loading fitting 14, outlet valve 15, coolant supply fittings 16.
According to the structural scheme in Figure 1b, a planetary actuator is designed and manufactured ( Figure 8). The design of the mechanism consists of the rack 1, an input shaft 2, a carrier 3, an output shaft 4, a sun wheel 5, elliptical gears 6 and 7 on the output shaft, located at an angle of 180 ∘ , a first satellite consisting of a cylindrical gear 8, an elliptical gear 9 and a shaft 10, a second satellite consisting of a counterweight 11, an elliptical wheel 12 and a shaft 13.
The links of the mechanism have the following inertial characteristics (the numbers of the links correspond to A set of replaceable elliptical gears with eccentricities from 0.28 to 0.6 is manufactured for the planetary mechanism. Kinematic analysis of the mechanism is carried out in [14] by constructing plans of the link velocities for various eccentricities of elliptical gears, while the largest angle of rotation of the output shaft 147 ∘ is obtained at e = 0.6. The maximum angular speed of the motor shaft is 62.8 rad/s (n = 600 rpm), which allows to obtain the frequency up to 10 Hz of the rotationally reciprocating motion of impeller. Since the loads on the links of the mechanism increase with an increase in the frequency and angle of oscillation of the impeller, the kinetostatic analysis is carried out for the most loaded mode α = 150 ∘ , f = 10 Hz.
Studies of fluid dynamics in a reactor with a rotationally reciprocating motion of the impeller were carried out by various authors [35][36][37], equations for determining the Reynolds number and the moment of resistance on the working body were obtained. At the same time, researchers agree that in the case of laminar fluid motion, the moment of resistance is proportional to the angular velocity of the impeller, and in turbulent motion, it is proportional to the square of the angular velocity. In the selected operating mode of the stirred tank, a turbulent mode of fluid movement is observed, therefore, according to [37], the following equation is applied to determine the moment of resistance Mr on the impeller: where B is the coefficient of resistance square law, ω im is the angular velocity of the impeller. According to the results of calculations for the selected mode, B = 2.67 · 10 −6 is obtained and used in further calculations. After substituting the initial data and solving equations (2) to (13), the reactions in kinematic pairs is determined ( Figure 9). Studies have shown that the most loaded are gears (kinematic pairs) B and F, and the rotational pair G. Figure 9 presents graphs of the following reaction forces in kinematic pairs: -functions of forces R x 41 (φ 1 ), R y 41 (φ 1 ) in kinematic pair D and force R 43 (φ 1 ) in kinematic pair E are shown in Figure 9a; -functions of forces R x 30 (φ 1 ), R y 30 (φ 1 ) in kinematic pair G and force R 32 (φ 1 ) in kinematic pair F are shown in Figure 9b; -functions of forces R x 21 (φ 1 ), R y 21 (φ 1 ) in kinematic pair C and force R 20 (φ 1 ) in kinematic pair B are shown in Figure 9c; -functions of forces R x 10 (φ 1 ), R y 10 (φ 1 ) in kinematic pair A and balance moment M b (φ 1 ) are shown in Figure 9d.
As it can be seen from the graphs in Figure 9, the maximum values of the forces in the kinematic pairs and the balancing moment are observed during a change in the direction of rotation of the impeller (link 3). Moreover, in most cases, there is a change in the direction of the force and moment vector.

Conclusions
So, in this work, there are obtained systems of equations that allow to perform force analysis of the planetary actuator with elliptical gears and determine the reactions in kinematic pairs, as well as the balancing moment for all the positions of the mechanism. As an example, the calculations of the rotationally reciprocating stirred tank are carried out and the force graphs in kinematic pairs are presented, depending on the position of the mechanism, for the mode f = 10 Hz, α = 150 ∘ . The proposed methodology of force analysis can be implemented for other operating modes of the actuator, which allows for extensive studies of kinetostatics, to choose the most rational modes of operation of the stirred tank, as well as to simplify the calculation and design of the future machine. Also, the conducted studies can be applied in the design and analysis of other devices with the proposed scheme of the planetary mechanism.